Criteria for exponential dichotomy for triangular systems

We study the exponential dichotomy properties of nonautonomous systems of linear differential equations. Any such system is kinematically similar to a triangular system. Since it is easy to determine whether or not a diagonal system has an exponential dichotomy, it is important to study the relation between the exponential dichotomy properties of the triangular system and its diagonal part. Without loss of generality, we consider block upper triangular systems and study the relation with their diagonal parts. Our study addresses both bounded and unbounded systems, on the whole line and a half line. We conclude with a study of the spectral properties of bounded systems, focusing on the relation between the whole line spectrum and the half line spectra.